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Vector Equations of Lines in 3D

Key Terms

Vector form
r = a + λd, where a is a position vector of a known point on the line, d is the direction vector, and λ ∈ ℝ is the parameter.
Parametric form
x = a₁ + λd₁,   y = a₂ + λd₂,   z = a₃ + λd₃
Cartesian (symmetric) form
(x − a₁)/d₁ = (y − a₂)/d₂ = (z − a₃)/d₃ (valid when all components of d are non-zero).
Parallel lines
Two lines are parallel if their direction vectors are proportional (scalar multiples). Parallel lines with different positions do not intersect.
Intersecting lines
Solve the parametric equations simultaneously for λ and μ. Check all three equations are consistent.
Skew lines
Lines that are neither parallel nor intersecting. This can only happen in 3D (not 2D).
Distance between skew lines
d = |(b − a) · (d₁ × d₂)| / |d₁ × d₂|, where a, b are points on each line and d₁, d₂ are their direction vectors.
Key Formulas:
  • Vector equation: r = a + λd
  • Parametric: x = a₁+λd₁, y = a₂+λd₂, z = a₃+λd₃
  • Cartesian: (x−a₁)/d₁ = (y−a₂)/d₂ = (z−a₃)/d₃
  • Distance between skew lines: |(b−a)·(d₁×d₂)| / |d₁×d₂|
Hot Tip To test if two lines intersect: set up the parametric equations equal to each other (using parameters λ and μ), form a system of 3 equations in 2 unknowns. Solve any two equations for λ and μ, then substitute into the third to check consistency. If inconsistent, the lines are skew (assuming they are not parallel).

Worked Example 1 — Writing the equation of a line

Find the vector, parametric and Cartesian equations of the line through A = (2, −1, 3) with direction d = (1, 2, −2).

Vector:   r = (2, −1, 3) + λ(1, 2, −2)

Parametric:   x = 2 + λ,   y = −1 + 2λ,   z = 3 − 2λ

Cartesian:   (x − 2)/1 = (y + 1)/2 = (z − 3)/(−2)

Worked Example 2 — Classifying two lines

Line L₁: r = (1,0,2) + λ(1,1,−1).   Line L₂: r = (2,1,0) + μ(2,2,−2).

Direction vectors: (1,1,−1) and (2,2,−2) = 2(1,1,−1). They are proportional, so the lines are parallel.

Check if they are the same line: Is (2,1,0) on L₁? Need (2,1,0) = (1,0,2) + λ(1,1,−1), giving λ=1 from x, λ=1 from y, but z: 2−1=1≠0. Different lines, so parallel and distinct.

Deriving the Vector Equation of a Line

A line in 3D is uniquely determined by a point on it and a direction. If point A has position vector a and the line runs in direction d, then any point P on the line satisfies OP = a + λd for some real number λ. As λ varies over all reals, P traces out the entire line. The vector d is called the direction vector.

The key insight: two points A and B determine a line, with direction d = B − A (or any scalar multiple). A line through A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) has vector equation r = (a₁, a₂, a₃) + λ(b₁−a₁, b₂−a₂, b₃−a₃).

Converting Between Forms

The three forms are equivalent and can be converted:

  • From vector to parametric: write each component separately.
  • From parametric to Cartesian: solve each for λ and equate.
  • From Cartesian to vector/parametric: read off a and d directly.

If a direction component is zero (e.g. d₂ = 0), the Cartesian form cannot be written with a fraction in that component. Instead, write d₂ = 0 as a separate equation: y = a₂. Then write the remaining ratios as (x−a₁)/d₁ = (z−a₃)/d₃.

Classifying Pairs of Lines in 3D

Given two lines L₁: r = a + λd₁ and L₂: r = b + μd₂:

  • Parallel: d₁ = kd₂ for some scalar k, and the lines have different positions.
  • Identical: d₁ = kd₂ and a point of one lies on the other.
  • Intersecting: Not parallel, and there exist λ, μ satisfying a + λd₁ = b + μd₂.
  • Skew: Not parallel, and no λ, μ satisfy the simultaneous equations. Skew lines can only exist in 3D — in 2D any two non-parallel lines must intersect.

Distance Between Skew Lines

The shortest distance between skew lines L₁ and L₂ is measured along the unique common perpendicular. If n = d₁ × d₂ is the cross product of the direction vectors, then n is perpendicular to both lines. The distance is the length of the projection of b − a onto n:

d = |(b − a) · n| / |n|

where a is a point on L₁ and b is a point on L₂.

Exam Tip: When testing for intersection, always check your solution in the third equation — the system is overdetermined (3 equations, 2 unknowns), and the third equation is the consistency check.
Exam Tip: The direction vector of a line is not unique — any non-zero scalar multiple gives the same line. Always simplify your direction vector (e.g. reduce (2, 4, −6) to (1, 2, −3)) for cleaner working.

Mastery Practice

See Answers ➔

  1. Write the vector equation of the line passing through the given point with the given direction. Fluency

    1. (a) Point (1, 2, 3), direction (2, −1, 4)
    2. (b) Point (0, −1, 5), direction (1, 3, −2)

  2. Write the parametric equations for the line through A = (3, 0, −2) and B = (1, 4, 1). Fluency

  3. Convert to Cartesian (symmetric) form: r = (2, −3, 1) + λ(3, 2, −1). Fluency

  4. Determine whether the point P lies on the given line. Fluency

    1. (a) P = (5, −1, 7), line r = (1, 3, −1) + λ(2, −2, 4)
    2. (b) P = (3, 2, 0), line r = (1, 1, 2) + λ(1, 0, −1)

  5. Find the point of intersection of the lines L₁: r = (1, 2, 0) + λ(1, −1, 2) and L₂: r = (3, 0, 4) + μ(−1, 1, 0). Understanding

  6. Classify each pair of lines as parallel, intersecting, or skew. Understanding

    1. (a) L₁: r = (0,1,2)+λ(1,2,1),   L₂: r = (1,3,4)+μ(2,4,2)
    2. (b) L₁: r = (1,0,0)+λ(1,1,0),   L₂: r = (0,1,1)+μ(1,0,1)

  7. Find the distance between the skew lines L₁: r = (1,0,0)+λ(1,1,0) and L₂: r = (0,1,0)+μ(0,1,1). Understanding

  8. A line passes through P = (2, −1, 4) and Q = (5, 3, −2). Find: (a) the vector equation, (b) whether the point R = (8, 7, −8) lies on the line. Understanding

  9. Find the distance between the skew lines L₁: r = (0,0,0)+λ(1,0,1) and L₂: r = (1,1,0)+μ(0,1,1). Problem Solving

  10. Two aircraft fly along the paths L₁: r = (10, 0, 5) + λ(2, 1, −1) and L₂: r = (0, 5, 7) + μ(1, −1, 0) (distances in km). Show that the flight paths are skew, and find the minimum distance between the paths. Problem Solving